# A brief survey on discontinuous Galerkin methods in computational ﬂuid dynamics ∗ ```力学进展 2013 年 第 43 卷 第 6 期: 541-553
A brief survey on discontinuous Galerkin
methods in computational ﬂuid dynamics∗
Chi-Wang Shu†
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Abstract
Discontinuous Galerkin (DG) methods combine features in ﬁnite element methods
(weak formulation, ﬁnite dimensional solution and test function spaces) and in ﬁnite volume
methods (numerical ﬂuxes, nonlinear limiters) and are particularly suitable for simulating convection dominated problems, such as linear and nonlinear waves including shock waves. In
this article we will give a brief survey of DG methods, emphasizing their applications in computational ﬂuid dynamics (CFD). We will discuss essential ingredients and properties of DG
methods, and will also give a few examples of recent developments of DG methods which have
facilitated their applications in CFD.
Keywords: discontinuous Galerkin methods, computational ﬂuid dynamics,
Classification code: O35
Document code: A
1 Introduction
Discontinuous Galerkin (DG) methods belong
DOI: 10.6052/1000-0992-13-059
leads to ambiguities at element interfaces, the
technique from ﬁnite volume methodology, namely
to the class of ﬁnite element methods. They are
the choice of numerical ﬂuxes, is introduced into
based on weak formulations and with ﬁnite dimen-
the DG schemes. Another important technique
sional piecewise polynomial solution space and test
from the ﬁnite volume methodology, namely the
function space. The main diﬀerence with tradi-
choice of nonlinear limiters to control spurious os-
tional ﬁnite element methods is that the ﬁnite elecillations in the presence of strong discontinuities,
ment function space corresponding to DG methods
consists of piecewise polynomials (or other simple
is also introduced into the DG schemes. From this
functions) which are allowed to be completely dis-
point of view, DG schemes can be considered as
continuous across element interfaces. Since this
hybrid ﬁnite element and ﬁnite volume schemes.
∗
This work was partially supported by US NSF grant DMS-1112700 and by the Open Fund of State Key Laboratory of
High-temperature Gas Dynamics, China (No. 2011KF02).
†
E-mail: shu@dam.brown.edu
Cite this paper: Chi-Wang Shu. A brief survey on discontinuous Galerkin methods in computational ﬂuid dynamics.
Advances in Mechanics, 2013, 43(6): 541–553
542

2013 年 第 43 卷

The ﬁrst DG method was introduced in 1973
and weather forecasting, among many others. For
by Reed and Hill in a Los Alamos technical report
earlier work on DG methods, we refer to the sur-
(Read and Hill 1973). It solves the equations for
vey paper (Cockburn et al. 2000), and other pa-
neutron transport, which are time independent lin-
pers in that Springer volume, which contains the
ear hyperbolic equations. A major development of
conference proceedings of the First International
the DG method was carried out by Cockburn et al.
Symposium on Discontinuous Galerkin Methods
in a series of papers (Cockburn et al. 1989a, 1989b,
held at Newport, Rhode Island in 1999. The lec-
1990, 1991, 1998), in which the authors have estab-
ture notes (Cockburn 1999) is a good reference for
lished a framework to easily solve nonlinear time
many details, as well as the extensive review pa-
dependent hyperbolic equations, such as the Euler
per (Cockburn and Shu 2001). The review paper
equations of compressible gas dynamics. The DG
(Xu and Shu 2010) covers the local DG method
method of Cockburn et al. belongs to the class
for partial diﬀerential equations (PDEs) contain-
of method-of-lines, namely the DG discretization
ing higher order spatial derivatives, such as Navier-
is used only for the spatial variables, and explicit,
Stokes equations. More recently, there are three
nonlinearly stable high order Runge-Kutta meth-
special journal issues devoted to the DG method
ods (Shu and Osher 1988, Gottlieb et al. 2011)
(Cockburn et al. 2005, 2009, Dawson 2006), which
are used to discretize the time variable. The two
contain many interesting papers on DG method
techniques from the ﬁnite volume methodology
in all aspects including algorithm design, analysis,
mentioned above, namely the usage of exact or
implementation and applications. There are also
approximate Riemann solvers as interface ﬂuxes
a few recent books and lecture notes (Hesthaven
and total variation bounded (TVB) nonlinear lim-
et al. 2008, Kanschat 2007, Li 2006, Riviere 2008,
iters (Shu 1987) to achieve non-oscillatory prop-
Shu 2009) on DG methods.
erties for strong shocks, were also introduced into
the DG method of Cockburn et al. At the beginning, applications of DG methods to computational ﬂuid dynamics (CFD) were mainly for
solving Euler equations of compressible gas dy-
2 DG methods for hyperbolic conservation laws
In this section, we describe brieﬂy the main
idea of DG methods for solving hyperbolic conser-
namics. Later, the DG methodology was gener-
vation laws. In CFD such equations include Eu-
alized to treat viscous terms as well and hence
ler equations of compressible gas dynamics, MHD
the DG schemes were designed to solve Navier-
equations, linearized Euler equations in aeroacous-
Stokes equations (Bassi and Rebay 1997, Cock-
tics, shallow water equations, etc. We use the fol-
burn and Shu 1998). Within the umbrella of CFD,
lowing one-dimensional scalar equation
the DG methods have also been applied to areas
ut + f (u)x = 0
(1)
including aeroacoustics, granular ﬂows, magnetohydrodynamics, meteorology, modeling of shallow
as an example.
water, oceanography, transport of contaminant in
Let us ﬁrst settle on some notations. Assum-
porous media, turbulent ﬂows, viscoelastic ﬂows
ing we are solving (1) in x ∈ [0, 1]. We divide [0,1]
Chi-Wang Shu : A brief survey on discontinuous Galerkin methods &middot; &middot; &middot;

543
Consistency: fˆ(u, u) = f (u);
into N cells
Continuity: fˆ(u− , u+ ) is at least Lipschitz
0 = x 12 &lt; x 32 &lt; &middot; &middot; &middot; &lt; xN + 12 = 1,
continuous with respect to both arguments u− and
u+ .
and denote
(
)
Ij = xj− 12 , xj+ 12 ,
xj =
)
1(
xj− 12 + xj+ 12 ,
2
hj = xj+ 12 − xj− 12
Monotonicity: fˆ(u− , u+ ) is a non-decreasing
function of its ﬁrst argument u− and a nonincreasing function of its second argument u+ .
as the cells, cell centers and cell lengths respectively. We also deﬁne h = hmax = maxj hj and
hmin = minj hj , and we consider only regular
meshes, that is hmax 6 λhmin where λ &gt; 1 is a
constant during mesh reﬁnement. Deﬁne the discontinuous Galerkin ﬁnite element space as
Symbolically fˆ(↑, ↓).
We refer to, e.g., (LeVeque 1990) for more
Notice that the DG scheme Eq. (3) can also
be written as
∫
((uh )t + f (uh )x ) vh dx +
Ij
Vhk
= {v : v|Ij ∈ P (Ij ), j = 1, &middot; &middot; &middot; , N },
k
(2)
(fˆj+ 12 − f ((uh )−
)(vh )−
−
j+ 1
j+ 1
2
where P k (Ij ) denotes the space of polynomials in
Ij of degree at most k. This polynomial degree k
can actually change from cell to cell (p-adaptivity),
simplicity.
2
(fˆj− 21 − f ((uh )+
)(vh )+
=0
j− 1
j− 1
2
(5)
2
through integration by parts. Mathematically the
two formulations Eq. (3) and Eq. (5) are equivalent, so users can choose to implement any one of
them and will get the same result. However, if the
The semi-discrete DG method for solving (1)
is deﬁned as follows: ﬁnd the unique function
uh = uh (t) ∈ Vhk such that, for all test functions
vh ∈ Vhk and all 1 6 j 6 N , we have
∫
∫
(uh )t vh dx −
f (uh )(vh )x dx +
Ij
2
mated by numerical quadrature rules, then the two
formulations may no longer be equivalent. Some of
the variants of DG schemes, for example the CPR
scheme (Wang and Gao 2009), can be considered
Ij
fˆj+ 12 (vh )−
− fˆj− 12 (vh )+
= 0.
j+ 1
j− 1
integral terms in Eq. (3) and Eq. (5) are approxi-
(3)
as scheme (5) with the integral term replaced by
2
Here, fˆi+ 12 is the numerical ﬂux, which is a single-
If Eq. (1) is a system of hyperbolic conserva-
valued function deﬁned at the cell interfaces and
tion laws, the formulation of the DG scheme is the
in general depends on the values of the numerical
same as in the scalar case, except that the numer-
solution uh from both sides of the interface
ical ﬂux Eq. (4) is no longer a monotone ﬂux but
, t), uh (x+
, t)).
fˆi+ 12 = fˆ(uh (x−
i+ 1
i+ 1
2
a ﬂux based on an exact or approximate Riemann
(4)
2
solver (Toro 1991).
We use the so-called monotone ﬂuxes from ﬁnite
Since the DG scheme is deﬁned in a weak
volume schemes, which satisfy the following con-
form, its multi-dimensional version is similar to
ditions:
its one-dimensional version, with integration by

544

parts replaced by Green’s formulas. The numerical

2013 年 第 43 卷

3 DG
methods
for
convection-
ﬂuxes are still one-dimensional, along the normal
diﬀusion equations
direction of element interfaces. Elements can be
In this section we describe brieﬂy the main
of any shape, and mixed type meshes (e.g. both
idea of DG methods for solving convection-
triangles and quadrilateral cells in 2D) can be eas-
diﬀusion equations.
ily accommodated. The method is thus easy to
include Navier-Stokes equations of compressible
handle complicated geometry and boundary con-
gas dynamics, etc.
ditions.
dimensional scalar equation
Time discretization is usually through ex-
In CFD such equations
We use the following one-
ut + f (u)x = (a(u)ux )x
(6)
plicit Runge-Kutta methods, for example the
total-variational-diminishing (TVD), or strong-
with a(u) &gt; 0 as an example. We ﬁrst discuss
stability-preserving (SSP) time discretizations
the local DG (LDG) method (Cockburn and Shu
(Shu and Osher 1988, Gottlieb et al. 2011). The
1998), for which we rewrite Eq. (6) as the following
fully discretized scheme is referred to as Runge-
system
Kutta DG (RKDG) schemes.
The RKDG schemes are already energy sta-
ut + f (u)x = (b(u)q)x , q − B(u)x = 0,
where
ble (Jiang and Shu 1994, Hou and Liu 2007, Zhang
b(u) =
and Shu 2010), and can be used to solve Eq. (1)
∫
√
a(u), B(u) =
(7)
u
b(u)du.
(8)
with smooth solutions or solutions with only weak
The ﬁnite element space is still given by Eq. (2).
discontinuities. However, for solutions with strong
The semi-discrete LDG scheme is deﬁned as fol-
shocks, DG schemes will generate spurious oscilla-
lows. Find uh , qh ∈ Vhk such that, for all test func-
tions which may lead to nonlinear instability and
tions vh , ph ∈ Vhk and all 1 6 i 6 N , we have
∫
∫
(uh )t (vh )dx − (f (uh ) −
blow-ups of the code. In such cases, some form of
nonlinear limiters would be needed. We will ad-
Ii
b(uh )qh )(vh )x dx + (fˆ − b̂q̂)i+ 12 (vh )−
−
i+ 1
dress this issue in Section 4.
The RKDG method is local, with communications only with immediate neighbors through the
numerical ﬂuxes, regardless of the order of accuracy of the scheme. This makes the implemen-
Ii
2
(fˆ − b̂q̂)i− 21 (vh )+
= 0,
i− 21
∫
∫
qh ph dx +
B(uh )(ph )x dx −
Ii
Ii
−
B̂i+ 12 (ph )i+ 1
2
(9)
+ B̂i− 12 (ph )+
= 0.
i− 1
2
tation of the RKDG method highly eﬃcient. It
Here, all the “hat” terms are the numerical ﬂuxes.
also makes the method easy for parallel imple-
We already know from Section 2 that the convec-
mentation. The method can achieves almost 100%
tion ﬂux fˆ should be chosen as a monotone ﬂux.
parallel eﬃciency for static meshes and over 80%
However, the upwinding principle is no longer a
parallel eﬃciency for dynamic load balancing with
valid guiding principle for the design of the diﬀu-
adaptive meshes (Biswas et al. 1994, Remacle et
sion ﬂuxes b̂, q̂ and B̂. According to Cockburn and
al. 2003). The DG method is also very friendly to
Shu (1998), suﬃcient conditions for the choices of
the GPU environment (Klockner et al. 2010).
these diﬀusion ﬂuxes to guarantee the stability of

Chi-Wang Shu : A brief survey on discontinuous Galerkin methods &middot; &middot; &middot;
545
the scheme Eq. (9) are given. Here, we will only
for solving problems with strong discontinuities,
discuss a particularly attractive choice, called “al-
the DG solution may generate spurious numerical
ternating ﬂuxes”, deﬁned as
oscillations. In practice, especially for nonlinear
−
B(u+
h ) − B(uh )
, q̂ = qh+ , B̂ = B(u−
h ). (10)
−
−
u
u+
h
h
problems containing strong shocks, we often need
The important point is that q̂ and B̂ should be
tions. Most of the limiters studied in the literature
chosen from diﬀerent directions. Thus, the choice
come from the methodologies of ﬁnite volume high
b̂ =
b̂ =
−
B(u+
h ) − B(uh )
, q̂ = qh− , B̂ = B(u+
h)
−
u+
−
u
h
h
is also ﬁne.
to apply nonlinear limiters to control these oscilla-
resolution schemes.
A limiter can be considered as a postprocessor of the computed DG solution. In any
Notice that, from the second equation in the
cell which is deemed to contain a possible discon-
scheme (9), we can solve qh explicitly and locally
tinuity (the so-called troubled cells), the DG poly-
(in cell Ii ) in terms of uh , by inverting the small
nomial is replaced by a new polynomial of the same
mass matrix inside the cell Ii . This is why the
degree, while maintaining the original cell average
method is referred to as the “local” discontinuous
for conservation. Diﬀerent limiters compute this
Galerkin method (LDG).
new polynomial in diﬀerent fashions. The main
LDG method as deﬁned above is energy stable
idea is to require that the new polynomial is less
(Cockburn and Shu 1998). Its multi-dimensional
oscillatory than the old one, and, if the solution in
version is still similar to its one-dimensional ver-
this cell happens to be smooth, then the new poly-
sion, and the advantages in local communication,
nomial should have the same high order accuracy
easiness in handling complicated geometry and
as the old one. Some of the limiters are applied to
boundary conditions, and parallel eﬃciency, are
all cells, while they should take eﬀect (change the
still valid.
polynomial in the cell) only in the cells near the
There are other types of DG approxima-
discontinuities. The total variation diminishing
tions to the viscous terms (Bassi and Rebay 1997,
(TVD) limiters (Harten 1983) belong to this class.
Wheeler 1978, Arnold 1982, Baumann and Oden
Unfortunately, such limiters tend to take eﬀect
1999, Oden et al. 1998, Liu and Yan 2009, 2010,
also in some cells in which the solution is smooth,
Cheng and Shu 2008).
for example in cells near smooth extrema of the
exact solution. Accuracy is therefore lost in such
4 Examples of recent developments
on DG methods
(Shu 1987), applied to RKDG schemes (Cockburn
In this section we give a few examples of recent developments on DG methods which are relevant to CFD.
4.1
et al. 1989, Cockburn et al. 1989, Cockburn et
al. 1990, Cockburn and Shu 1998), attempt to
remove this diﬃculty and to ensure that the lim-
Nonlinear limiters
iter takes eﬀect only on cells near the discontinu-
The RKDG schemes for conservation laws deﬁned in Section 2 are energy stable.
cells. The total variation bounded (TVB) limiters
However,
ities. The TVB limiters are widely used in applications, because of their simplicity in implementa-

546

2013 年 第 43 卷
tion. However, the TVB limiters involve a param-
formation from not only the immediate neighbor-
eter M , related to the value of the second deriva-
ing cells but also neighbors’ neighbors, making it
tive of the exact solution near smooth extrema,
complicated to implement in multi-dimensions, es-
which must be chosen by the user for diﬀerent test
pecially for unstructured meshes (Zhu et al. 2008,
cases. The moment-based limiter (Biswas et al.
Hu and Shu 1999, Zhang and Shu 2009). It also
1994) and the improved moment limiter (Burbeau
destroys the local data structure of the base DG
et al. 2001) also belong to this class, and they are
scheme (which needs only to communicate with
speciﬁcally designed for DG methods and limit the
immediate neighbors). The eﬀort made by Qiu
moments of the polynomial sequentially, from the
and Shu (2003, 2005) attempts to construct Her-
highest order moment downwards. Unfortunately,
mite type WENO approximations, which use the
the moment-based limiters may also take eﬀect in
information of not only the cell averages but also
certain smooth cells, thereby destroying accuracy
the lower order moments such as slopes, to reduce
in these cells.
the spread of reconstruction stencils. However, for
The limiters based on the weighted essentially non-oscillatory (WENO) methodology are
higher order methods, the information of neighbors’ neighbors is still needed.
designed with the objective of maintaining the
More recently, Zhong and Shu (2012) devel-
high order accuracy even if they take eﬀect in
oped a new WENO limiting procedure for RKDG
smooth cells.
These limiters are based on the
methods on structured meshes. The idea is to re-
WENO methodology for ﬁnite volume schemes
construct the entire polynomial, instead of recon-
(Liu et al. 1994, Jiang and Shu 1996), and in-
structing point values or moments in the classical
volve nonlinear reconstructions of the polynomials
WENO reconstructions. That is, the entire recon-
in troubled cells using the information of neigh-
struction polynomial on the target cell is a con-
boring cells. The WENO reconstructed polynomi-
vex combination of polynomials on this cell and
als have the same high order of accuracy as the
its immediate neighboring cells, with suitable ad-
original polynomials when the solution is smooth,
justments for conservation and with the nonlinear
and they are (essentially) non-oscillatory near dis-
weights of the convex combination following the
continuities. Qiu and Shu (2005) and Zhu et al.
classical WENO procedure. The main advantage
(2008) designed WENO limiters using the usual
of this limiter is its simplicity in implementation,
WENO reconstruction based on cell averages of
as it uses only the information from immediate
neighboring cells as in (Jiang and Shu 1996, Hu
neighbors and the linear weights are always posi-
and Shu 1999, Shi et al. 2002), to reconstruct the
tive. This simplicity is more prominent for multi-
values of the solutions at certain Gaussian quadra-
dimensional unstructured meshes, which is studied
ture points in the target cells, and then rebuild the
in (Zhu et al. 2013) for two-dimensional unstruc-
solution polynomials from the original cell aver-
tured triangular meshes. This new WENO limiter
age and the reconstructed values at the Gaussian
has also been designed for CPR type schemes (Du
quadrature points through a numerical integration
et al.).
for the moments. This limiter needs to use the in-
The WENO limiters are typically applied

Chi-Wang Shu : A brief survey on discontinuous Galerkin methods &middot; &middot; &middot;
547
only in designated “troubled cells”. In order to
to preserve strict bounds (maximum principle for
save computational cost and to minimize the in-
scalar problems and positivity of relevant quanti-
ﬂuence of accuracy in smooth regions, a troubled
ties for scalar problems or systems), while main-
cell indicator is needed, to correctly identify cells
taining provable high order accuracy of the orig-
near discontinuities in which the limiters should
inal schemes. These techniques apply to multi-
be applied. Qiu and Shu (2005) have compared
dimensions in general unstructured triangulations
several troubled cell indicators. In practice, the
as well (Zhang and Shu 2010a, 2010b, Zhang et
TVB indicator (Shu 1987) and the KXRCF indi-
al., 2012).
cator (Krivodonova et al. 2004) are often the best
choices.
4.2
Bound preserving DG schemes
We will not repeat here the details of this general framework and refer the readers to the references. We will summarize here the main steps in
this framework:
In many CFD problems, the physical quantities have desired bounds which are satisﬁed by the
exact solutions of the PDEs. For example, for twodimensional incompressible Euler or Navier-Stokes
equations written in a vorticity-stream function
formulation, the vorticity satisﬁes a maximum
principle. For Euler equations of compressible gas
dynamics, density and pressure remain positive
(non-negative) when their initial values are positive. It would certainly be desirable if numerical
solutions obey the same bounds. If the numerical
solution goes out of the bounds because of spurious oscillations, it would either be non-physical
(e.g. negative density, negative internal energy, a
percentage of a component which goes below zero
or above one), or worse still, it could lead to nonlinear instability and blowups of the code because the
PDE becomes ill-posed (e.g. the Euler equations
of compressible gas dynamics become ill-posed for
negative density or pressure).
Not all limiters discussed in the previous section can enforce the bound-preserving property.
When they do, they often degenerate the order of
1. We ﬁrst ﬁnd a ﬁrst order base DG scheme,
using piecewise polynomials of degree zero (piecewise constants), which can be proved to be boundpreserving under certain Courant-Friedrichs-Lewy
(CFL) conditions for Euler forward time discretization. Notice that a ﬁrst order DG scheme
is same as a ﬁrst order ﬁnite volume scheme.
For scalar hyperbolic conservation laws in Eq.
(1), the ﬁrst order DG scheme using any monotone
numerical ﬂux would satisfy a maximum principle.
For Euler equations of compressible gas dynamics,
several ﬁrst order schemes, including the Godunov
scheme (Einfeldt et al. 1991), the Lax-Friedrichs
scheme (Perthame and Shu 1996, Zhang and Shu
2010), the Harten-Lax-van Leer (HLLE) (Harten
et al.
1983) scheme, and the Boltzmann type
kinetic scheme (Perthame 1992), are positivitypreserving for density and pressure.
2. We then apply a simple scaling limiter to
the high order DG solution at time level n. If the
DG solution at time level n in cell Ij is a polynomial pj (x), we replace it by the limited polynomial
p̃j (x) deﬁned by
accuracy of the original scheme in smooth regions.
Recently, a general framework is established
p̃j (x) = θj (pj (x) − ūnj ) + ūnj

548

where ūnj is the cell average of pj (x), and
{
M − ūn j θj = min ,
Mj − ūnj }
m − ūn j ,1 ,
mj − ūnj 进
2013 年 第 43 卷

and pressure (or internal energy) have been designed in Zhang and Shu (2010b, 2011a, 2011b,
2012) and in Zhang, Xia and Shu (2012).
For shallow water equations, high order
with
DG schemes maintaining non-negativity of water
Mj = max pj (x), mj = min pj (x)
x∈Sj
height have been designed in Xing, Zhang and Shu
x∈Sj
(2010).
where Sj is the set of certain Legendre Gauss-
Positivity-preserving semi-Lagrangian DG
Lobatto quadrature points of the cell Ij . Clearly,
schemes have been designed in Qiu and Shu (2011)
this limiter is just a simple scaling of the original
and in Rossmanith and Seal (2011).
polynomial around its average.
3.
We then evolve the solution by Euler
4.3
DG method for hyperbolic equations
involving δ-functions
forward time discretization, or by TVD or SSP
In a hyperbolic conservation law
Runge-Kutta time discretization (Shu and Osher
1988, Gottlieb et al. 2011).
We can see that this procedure is very simple
ut + f (u)x = g(x, t), (x, t) ∈ R &times; (0, T ],
u(x, 0) = u0 (x),
x ∈ R,
(11)
and inexpensive to implement. The scaling lim-
the initial condition u0 , or the source term g(x, t),
iter involves only evaluation of the DG polynomial
or the solution u(x, t) may contain δ-singularities.
at pre-determined quadrature points. The proce-
Such singularities are more diﬃcult to handle than
dure can be applied in arbitrary triangular meshes.
discontinuities in the solutions. Many high order
Amazingly, this simple process leads to mathemat-
schemes would easily blow up in the presence of δ-
ically provable bound-preserving property without
function singularities, because of the severe oscil-
degenerating the high order accuracy of the base
lations leading to non-physical regimes (e.g. neg-
DG scheme.
ative density). On the other hand, if one applies
For scalar nonlinear conservation laws, pas-
traditional limiters such as various slope limiters
sive convection in a divergence-free velocity ﬁeld,
to enforce stability, the resolution of the δ-function
and 2D incompressible Euler equations in the
singularities would be seriously deteriorated. Re-
vorticity-stream function formulation, high order
solution is also seriously aﬀected by other com-
DG schemes maintaining maximum principle have
monly used strategies such as moliﬁcations by an
been designed in Zhang and Shu (2010) and in
approximate Gaussian to smear out the δ-function.
Zhang, Xia and Shu (2012).
Since DG methods are based on weak formu-
For scalar nonlinear convection diﬀusion
lations, they can be designed directly to handle δ-
equations, second order DG schemes on unstruc-
function singularities. Recently, we have designed
tured triangulations maintaining maximum princi-
and analyzed DG schemes for solving linear and
ple have been designed in Zhang and Shu (2013).
nonlinear PDE models with δ-function singulari-
For Euler equations of gas dynamics, high or-
ties (Yang and Shu 2013, Yang et al. 2013). For
der DG schemes maintaining positivity of density
linear problems, we prove stability and high order

Chi-Wang Shu : A brief survey on discontinuous Galerkin methods &middot; &middot; &middot;
549
error estimates in negative norms when the DG
1999). At least in one dimension, the viscosity so-
method is applied, and propose post-processing
lution of the Hamilton-Jacobi Eq. (12) is equiva-
techniques to recover high order accuracy in strong
lent to the entropy solution of the conservation law
norms away from these δ-function singularities.
in Eq. (1), when we identify φx = u and H = f .
For nonlinear problems, such as Krause’s consen-
Therefore, a DG scheme for solving the conserva-
sus models (Canuto et al. 2012) and pressure-
tion law in Eq. (1) can be directly used to approxi-
less Euler equations (Chen and Liu 2003), an ad-
mate the derivative of the viscosity solution of the
equate design of bound preserving limiter, within
Hamilton-Jacobi Eq. (12). The missing degree of
the framework described in Section 4.2, to enforce
freedom, to recover φ from φx , can be determined
the physical bounds without compromising reso-
by
∫
lution of δ-function singularities is shown to be
((φh )t + H(uh )) dx = 0,
Ij
crucial. With such limiters, high resolution and
highly stable results can be obtained for such dif-
which is an evolution equation for the cell average of φ when the approximation to its derivative
ﬁcult nonlinear models.
φx = u is known. This algorithm is well deﬁned
4.4
DG method for Hamilton-Jacobi equations and nonlinear control
Time dependent Hamilton-Jacobi equations
take the form
for one dimension. Additional complications exist
for multi-dimensional cases, which can be handled
either by a least square procedure (Hu and Shu
1999) or by using locally curl-free DG spaces (Li
φt + H(φx1 , ..., φxd ) = 0, φ(x, 0) = φ0 (x), (12)
and Shu 2005) (these two approaches are mathematically equivalent (Li and Shu 2005)).
where H is a Lipschitz continuous function. H
Even though the DG schemes in (Hu and Shu
could also depend on φ, x and t in some appli-
1999, Li and Shu 2005) are successful in approx-
cations. Hamilton-Jacobi equations appear often
imating the Hamilton-Jacobi Eq. (12), it involves
in many applications.
Examples in CFD in-
rewriting it as a conservation law satisﬁed by the
clude front propagation, level set methods, multi-
derivatives of the solution φ. It is desirable to
material ﬂows, and nonlinear control.
design a DG method which solves directly the so-
Since the spatial derivative operator is inside
lution φ to the Hamilton-Jacobi Eq. (12). The
the nonlinear Hamiltonian H in Eq. (12), integra-
scheme of Cheng and Shu (2007) serves this pur-
tion by parts leading to a weak formulation for the
pose. It starts from the alternative formulation of
DG scheme cannot be directly performed. How-
DG schemes given in Eq. (5) which does not re-
ever, by exploring the strong relationship between
quire explicit integration by parts to the original
the Hamilton-Jacobi equations and conservation
PDE to write down the DG scheme. Further de-
laws, various formulations of DG methods have
velopment and application of this method to prob-
been designed in the literature.
lems in optimal control are given in (Bokanowski
The ﬁrst attempt to design a DG method was
based exactly on this relationship (Hu and Shu
et al. 2010, 2011, 2013). Excellent simulation
results
are
ob-tained
simulations in these
for
nonlinear
control

550

references.

2013 年 第 43 卷
Baumann C E, Oden J T. 1999. A discontinuous h-p ﬁ-
Another DG method which solves directly the
Hamilton-Jacobi Eq. (12) is that of Yan and Osher (2011). This method is motivated by the local
discontinuous Galerkin (LDG) method for solving
second order partial diﬀerential equations (Cockburn and Shu 1998).
nite element method for convection-diﬀusion problems.
Computer Methods in Applied Mechanics and Engineering, 175: 311-341.
Biswas R, Devine K D, Flaherty J. 1994. Parallel, adaptive
ﬁnite element methods for conservation laws. Applied
Numerical Mathematics, 14: 255-283.
Bokanowski O, Cheng Y, Shu C W. 2011. A discontinuous
Galerkin solver for front propagation. SIAM Journal on
Scientific Computing, 33: 923-938.
5 Concluding remarks
Bokanowski O, Cheng Y, Shu C W. A discontinuous
Galerkin scheme for front propagation with obsta-
In this short survey article we have given an
cles.
Numerische Mathematik, to appear.
DOI:
10.1007/s00211-013-0555-3.
overview of discontinuous Galerkin (DG) methods
Bokanowski O, Cheng Y, Shu C W. Convergence of dis-
applied to computational ﬂuid dynamics (CFD)
continuous Galerkin schemes for front propagation with
problems. A few representative examples of re-
obstacles. submitted to Mathematics of Computation.
cent developments are given to illustrate the vitality of this research area. The DG method has
many advantages for large scale computing, especially for massively parallel (exascale) computing
environments.
Future research in DG methods
might include study on reliable and eﬃcient error
Burbeau A, Sagaut P, Bruneau Ch H. 2001.
A problem-
independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. Journal of Computational
Physics, 169: 111-150.
Canuto C, Fagnani F, Tilli P. 2012. An Eulerian approach
to the analysis of Krause’s consensus models.
SIAM
Journal on Control and Optimization, 50: 243-265.
Chen G Q, Liu H. 2003. Formation of δ-shocks and vacuum
states in the vanishing pressure limit of solutions to the
indicators to guide h-p adaptivity, thus realizing
Euler equations for isentropic ﬂuids. SIAM Journal on
the full power of DG methods for such adaptiv-
Mathematical Analysis, 34: 925-938.
ity; more eﬃcient and reliable limiters for solutions
Cheng Y, Shu C W. 2007. A discontinuous Galerkin ﬁnite
with strong shocks, which do not aﬀect accuracy
equations. Journal of Computational Physics, 223: 398-
in smooth regions and do not aﬀect convergence
415.
to steady state solutions; and DG methods with
element method for directly solving the Hamilton-Jacobi
Cheng Y, Shu C W. 2008. A discontinuous Galerkin ﬁnite
element method for time dependent partial diﬀerential
non-polynomial basis functions for multiscale and
equations with higher order derivatives. Mathematics of
turbulence modeling applications. It can be ex-
Computation, 77: 699-730.
pected that such further research in DG methods
will lead to the expansion of their application in
CFD and other scientiﬁc and engineering areas.
Cockburn B. 1999. Discontinuous Galerkin Methods for
Convection-Dominated Problems.
Berlin, Heidelberg,
Springer 69-224.
Cockburn B, Hou S, Shu C W. 1990. The Runge-Kutta
local projection discontinuous Galerkin ﬁnite element
method for conservation laws IV: the multidimensional
References
case. Mathematics of Computation, 54: 545-581.
An interior penalty ﬁnite element
Cockburn B, Karniadakis G, Shu C W. 2000. The De-
method with discontinuous elements. SIAM Journal on
velopment of Discontinuous Galerkin Methods. Berlin,
Numerical Analysis, 39: 742-760.
Heidelberg, Springer, 3-50.
Arnold D N. 1982.
Bassi F, Rebay S. 1997. A high-order accurate discontin-
Cockburn B, Lin S Y, Shu C W. 1989. TVB Runge-Kutta
uous ﬁnite element method for the numerical solution
local projection discontinuous Galerkin ﬁnite element
of the compressible Navier-Stokes equations. Journal of
method for conservation laws III: one-dimensional sys-
Computational Physics, 131: 267-279.
tems. Journal of Computational Physics, 84: 90-113.

Chi-Wang Shu : A brief survey on discontinuous Galerkin methods &middot; &middot; &middot;
551
Cockburn B, Shu C W. 1989. TVB Runge-Kutta local pro-
Hu C, Shu C W. 1999. Weighted essentially non-oscillatory
jection discontinuous Galerkin ﬁnite element method for
schemes on triangular meshes. Journal of Computational
conservation laws II: general framework. Mathematics of
Physics, 150: 97-127.
Computation, 52: 411-435.
Hu C, Shu C W. 1999. A discontinuous Galerkin ﬁnite
Cockburn B, Shu C W. 1991. The Runge-Kutta local projection P 1 -discontinuous-Galerkin ﬁnite element method
for scalar conservation laws.
Mathematical Modelling
and Numerical Analysis (M 2 AN ), 25: 337-361.
Cockburn B, Shu C W. 1998. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. Journal of Computational Physics,
141: 199-224.
Cockburn B, Shu C W. 1998.
element method for Hamilton-Jacobi equations. SIAM
Journal on Scientific Computing, 21: 666-690.
Jiang G S, Shu C W. 1994. On cell entropy inequality for
discontinuous Galerkin methods. Mathematics of Computation, 62: 531-538.
Jiang G S, Shu C W. 1996.
Eﬃcient implementation
of weighted ENO schemes. Journal of Computational
Physics, 126: 202-228.
The local discontinu-
ous Galerkin method for time-dependent convectiondiﬀusion systems. SIAM Journal on Numerical Anal-
Kanschat G. 2007. Discontinuous Galerkin Methods for
Klockner A, Warburton T, Bridge J, Hesthaven J. 2010.
ysis, 35: 2440-2463.
Cockburn B, Shu C W. 2001. Runge-Kutta discontinuous
Galerkin methods for convection-dominated problems.
Journal of Scientific Computing, 16: 173-261.
Cockburn B, Shu C W. 2005. Foreword for the special issue
on discontinuous Galerkin method. Journal of Scientific
Computing, 22-23: 1-3.
Nodal discontinuous Galerkin methods on graphics processors. Journal of Computational Physics, 228: 78637882.
Krivodonova L, Xin J, Remacle J F, Chevaugeon N, Flaherty J E. 2004. Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation
laws. Applied Numerical Mathematics, 48: 323-338.
Cockburn B, Shu C W. 2009. Foreword for the special issue
on discontinuous Galerkin method. Journal of Scientific
LeVeque R J. 1990. Numerical Methods for Conservation
Laws, Birkhauser Verlag, Basel.
Computing, 40: 1-3.
Dawson C. 2006. Foreword for the special issue on discontinuous Galerkin method. Computer Methods in Applied
Mechanics and Engineering, 195: 3183.
Du J, Shu C W, Zhang M. 2013. A simple weighted essen-
Li B. 2006. Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Basel Birkhauser.
Li F, Shu C W. 2005.
Reinterpretation and simpliﬁed
implementation of a discontinuous Galerkin method for
tially non-oscillatory limiter for the correction procedure
Hamilton-Jacobi equations. Applied Mathematics Let-
via reconstruction (CPR) framework. submitted to Ap-
ters, 18: 1204-1209.
Liu H, Yan J. 2009.
plied Numerical Mathematics.
Einfeldt B, Munz C D, Roe P L, Sjögreen B. 1991. On
Godunov-Type methods near low densities. Journal of
Computational Physics, 92: 273-295.
Gottlieb S, Ketcheson D, Shu C W. 2011. Strong Stability
Preserving Runge-Kutta and Multistep Time Discretizations, Singapore World Scientiﬁc.
on Numerical Analysis, 47: 675-698.
Liu H, Yan J. 2010.
The direct discontinuous Galerkin
(DDG) methods for diﬀusion with interface corrections.
Communications in Computational Physics, 8: 541-564.
Liu X, Osher S, Chan T. 1994. Weighted essentially non-
Harten A. 1983. High resolution schemes for hyperbolic
conservation laws. Journal of Computational Physics,
49: 357-393.
oscillatory schemes. Journal of Computational Physics,
115: 200-212.
Oden J T, Babuvska I, Baumann C E. 1998. A discon-
Harten A, Lax P D, van Leer B. 1983. On upstream differencing and Godunov type schemes for hyperbolic conservation laws. SIAM Review, 25: 35-61.
Hesthaven J, Warburton T. 2008.
The direct discontinuous Galerkin
(DDG) methods for diﬀusion problems. SIAM Journal
Nodal Discontinuous
Galerkin Methods, New York Springer.
Hou S, Liu X D. 2007. Solutions of multi-dimensional hy-
tinuous hp ﬁnite element method for diﬀusion problems.
Journal of Computational Physics, 146: 491-519.
Perthame B. 1992. Second-order Boltzmann schemes for
compressible Euler equations in one and two space dimensions. SIAM Journal on Numerical Analysis, 29:
1-19.
perbolic systems of conservation laws by square entropy
Perthame B, Shu C W. 1996. On positivity preserving ﬁ-
condition satisfying discontinuous Galerkin method.
nite volume schemes for Euler equations. Numerische
Journal of Scientific Computing, 31: 127-151.
Mathematik, 73: 119-130.

552
Qiu J M, Shu C W. 2011.

Positivity preserving semi-

2013 年 第 43 卷

Wang Z J, Gao H. 2009.
A unifying lifting colloca-
Lagrangian discontinuous Galerkin formulation: theoret-
tion penalty formulation including the discontinuous
ical analysis and application to the Vlasov-Poisson sys-
Galerkin, spectral volume/diﬀerence methods for conser-
tem. Journal of Computational Physics, 230: 8386-8409.
vation laws on mixed grids. Journal of Computational
Qiu J X, Shu C W. 2003. Hermite WENO schemes and
Physics, 228: 8161-8186.
their application as limiters for Runge-Kutta discontinu-
Wheeler M F. 1978. An elliptic collocation-ﬁnite element
ous Galerkin method: one-dimensional case. Journal of
method with interior penalties. SIAM Journal on Nu-
Computational Physics, 193: 115-135.
merical Analysis, 15: 152-161.
Qiu J X, Shu C W. 2005. Hermite WENO schemes and
Xing Y, Zhang X, Shu C W. 2010. Positivity preserving
their application as limiters for Runge-Kutta discontinu-
high order well balanced discontinuous Galerkin meth-
ous Galerkin method II: two dimensional case. Comput-
ods for the shallow water equations. Advances in Water
ers and Fluids, 34: 642-663.
Resources, 33: 1476-1493.
Runge-Kutta discontinuous
Xu Y, Shu C W. 2010. Local discontinuous Galerkin meth-
Galerkin method using WENO limiters. SIAM Journal
ods for high-order time-dependent partial diﬀerential
on Scientific Computing, 26: 907-929.
equations. Communications in Computational Physics,
Qiu J X, Shu C W. 2005.
Qiu J X, Shu C W. 2005.
A comparison of troubled-
cell indicators for Runge-Kutta discontinuous Galerkin
7: 1-46.
Yan J, Osher S. 2011.
A local discontinuous Galerkin
methods using weighted essentially nonoscillatory lim-
method for directly solving Hamilton Jacobi equations.
iters. SIAM Journal on Scientific Computing, 27: 995-
Journal of Computational Physics, 230: 232-244.
Yang Y, Shu C W. 2013. Discontinuous Galerkin method
1013.
Reed W H, Hill T R. 1973. Triangular mesh methods for
the Neutron transport equation, Los Alamos Scientiﬁc
Laboratory Report LA-UR-73-479, Los Alamos, NM
Remacle J F, Flaherty J, Shephard M. 2003. An adaptive
discontinuous Galerkin technique with an orthogonal basis applied to Rayleigh-Taylor ﬂow instabilities. SIAM
hyperbolic
equations
involving
δ-singularities:
negative-order norm error estimates and applications.
Numerische Mathematik, 124: 753-781.
Yang Y, Wei D, Shu C W. 2013. Discontinuous Galerkin
method for Krause’s consensus models and pressureless
Euler equations. Journal of Computational Physics, 252:
109-127.
Review, 45: 53-72.
Rivière B. 2008. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Philadelphia: SIAM
Rossmanith J A, Seal D C. 2011.
for
Zhang Q, Shu C W. 2010. Stability analysis and a priori
error estimates to the third order explicit Runge-Kutta
discontinuous Galerkin method for scalar conservation
laws. SIAM Journal on Numerical Analysis, 48: 1038-
A positivity-
preserving high-order semi-Lagrangian discontinuous
Galerkin scheme for the Vlasov-Poisson equations. Journal of Computational Physics, 230: 6203-6232.
Shi J, Hu C, Shu C W. 2002. A technique of treating negative weights in WENO schemes. Journal of Computational Physics, 175: 108-127.
Shu C W. 1987. TVB uniformly high-order schemes for conservation laws. Mathematics of Computation, 49: 105121.
1063.
Zhang X, Shu C W. 2010.
On maximum-principle-
satisfying high order schemes for scalar conservation
laws.
Journal of Computational Physics, 229: 3091-
3120.
Zhang X, Shu C W. 2010. On positivity preserving high
order discontinuous Galerkin schemes for compressible
Euler equations on rectangular meshes. Journal of Computational Physics, 229: 8918-8934.
Zhang X, Shu C W. 2011a. Positivity-preserving high order
Shu C W. 2009. Discontinuous Galerkin methods: General
discontinuous Galerkin schemes for compressible Euler
approach and stability. Numerical Solutions of Partial
equations with source terms. Journal of Computational
Diﬀerential Equations, Advanced Courses in Mathemat-
Physics, 230: 1238-1248.
ics CRM Barcelona, 149-201.
Zhang X, Shu C W. 2011b. Maximum-principle-satisfying
Shu C W, Osher S. 1988. Eﬃcient implementation of essen-
and positivity-preserving high order schemes for conser-
tially non-oscillatory shock-capturing schemes. Journal
vation laws: Survey and new developments, in: Proceed-
of Computational Physics, 77: 439-471.
ings of the Royal Society A, 467: 2752-2776.
Toro E F. 1999. Riemann Solvers and Numerical Methods
for Fluid Dynamics, Berlin: Springer-Verlag.
Zhang X, Shu C W. 2012. A minimum entropy principle
of high order schemes for gas dynamics equations. Nu-

Chi-Wang Shu : A brief survey on discontinuous Galerkin methods &middot; &middot; &middot;
merische Mathematik, 121: 545-563.
553
Zhong X, Shu C W. 2012. A simple weighted essentially
Maximum-principle-
nonoscillatory limiter for Runge-Kutta discontinuous
satisfying and positivity-preserving high order discontin-
Galerkin methods. Journal of Computational Physics,
uous Galerkin schemes for conservation laws on triangu-
232: 397-415.
Zhang X, Xia Y, Shu C W. 2012.
lar meshes. Journal of Scientific Computing, 50: 29-62.
Zhu J, Qiu J X, Shu C W, Dumbser M. 2008. Runge-
Zhang Y, Zhang X, Shu C W. 2013. Maximum-principle-
Kutta discontinuous Galerkin method using WENO lim-
satisfying second order discontinuous Galerkin schemes
iters II: unstructured meshes. Journal of Computational
for convection-diﬀusion equations on triangular meshes.
Physics, 227: 4330-4353.
Journal of Computational Physics, 234: 295-316.
Zhu J, Zhong X, Shu C W, Qiu J X. 2013. Runge-Kutta dis-
Zhang Y T, Shu C W. 2009. Third order WENO scheme on
continuous Galerkin method using a new type of WENO
three dimensional tetrahedral meshes. Communications
limiters on unstructured meshes. Journal of Computa-
in Computational Physics, 5: 836-848.
tional Physics, 248: 200-220.
(editor in charge: Jiang Zonglin)

†
Division of Applied Mathematics Brown University Providence, RI 02912, USA

Chi-Wang Shu has been a faculty member with Brown University since 1987. He served as
the Chairman of the Division of Applied Mathematics from 1999 to 2005, and is now the
Theodore B. Stowell University Professor of Applied Mathematics. His research interest
includes high order ﬁnite diﬀerence, ﬁnite element and spectral methods for solving hyperbolic and other convection dominated partial diﬀerential equations, with applications to
areas such as computational ﬂuid dynamics, semi-conductor device simulations and computational cosmology. He is one of the major developers of several very popular high order
accurate schemes including weighted essentially non-oscillatory (WENO) ﬁnite diﬀerence
schemes, Runge-Kutta discontinuous Galerkin methods, and strong stability preserving
(SSP) time discretizations. He served as the Managing Editor of Mathematics of Computation between 2002
and 2012, is now the Chief Editor of Journal of Scientific Computing and the Co-Chief Editor of Methods and
Applications of Analysis, and he currently serves in the editorial boards of 14 other journals including Journal
of Computational Physics and Science China Mathematics. His honors include the First Feng Kang Prize of
Scientiﬁc Computing in 1995 and the SIAM/ACM Prize in Computational Science and Engineering in 2007. He
is an ISI Highly Cited Author in Mathematics, a SIAM Fellow in the inaugural class, and an AMS Fellow in the
inaugural class.
†
E-mail: shu@dam.brown.edu
```